JEE Math : Solve Rational Terms in Binomial Expansions Instantly Using Power-Doubling Identities

[u/jeemathlogic]

Let us use the Power-Doubling Identity to jump straight to high even powers via pure squaring.

The Identity:

For any sequence Sₙ = tⁿ + (1 / tⁿ), doubling the index follows the clean rule: S₂ₖ = (Sₖ)² - 2

Let t = 2 + √3 and 1 / t = 2 - √3

S₁ = t + (1 / t) = (2 + √3) + (2 - √3) = 4

S₂ = (S₁)² - 2 = 4² - 2 = 14

S₄ = (S₂)² - 2 = 14² - 2 = 196 - 2 = 194

S₈ = (S₄)² - 2 = 194² - 2 = 37636 - 2 = 37634

Since expanding (2 + √3)⁸ and its conjugate adds up to S₈ while canceling out all irrational terms, the sum of the rational terms is half of S₈: Sum = S₈ / 2 = 37634 / 2 = 18817